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Wavy Vortex Flow A tale of chaos, symmetry and serendipity in a steady world Greg King University of Warwick (UK) Collaborators: • Murray Rudman (CSIRO) • George Rowlands (Warwick) • Thanasis Yannacopoulos (Aegean) • Katie Coughlin (LLNL) • Igor Mezic (UCSB)
To understand this lecture you need to know • • • Some fluid dynamics Some Hamiltonian dynamics Something about phase space Poincare sections Need > 2 D phase space to get chaos Symmetry can reduce the dimensionality of phase space • Some knowledge of diffusion • A “friendly” applied mathematician !!
Dynamical Systems and Phase Space
Classical Mechanics and Phase Space Hamiltonian Dissipative
Fluid Dynamics and Phase Space 2 D incompressible fluid 3 D incompressible fluid Phase Space No chaos here Symmetries -- can reduce phase space
Poincare Sections (Experimental – i. e. , light sheet)
Eccentric Couette Flow Chaiken, Chevray, Tabor and Tan, Proc Roy Soc 1984 ? ? Illustrates “Significance” of KAM theory
Fountain et al, JFM 417, 265 -301 (2000) Stirring creates deformed vortex
Fountain et al, JFM 417, 265 -301 (2000) Experiment (light sheet) Numerical Particle Tracking (“light sheet”)
Taylor-Couette Radius Ratio: = a/b Reynolds Number: b a Re = a (b-a)/
Engineering Applications • • Chemical reactors Bioreactors Blood – Plasma separation etc
Taylor-Couette regime diagram (Andereck et al) Rein Reout
Some Possible Flows Taylor vortices Twisted vortices Wavy vortices Spiral vortices
Taylor Vortex Flow TVF -– Centrifugal instability of circular Couette flow. – Periodic cellular structure. – Three-dimensional, rotationally symmetric: u = u(r, z) Flat inflow and outflow boundaries are barriers to inter-vortex transport.
Rotational Symmetry 3 D 2 D Phase Space “Light Sheet” nested streamtubes /2 Z 0 inner cylinder Radius outer cylinder
Wavy Vortex Flow Taylor vortex flow wavy vortex flow Rec
The Leaky Transport Barrier Wavy vortex flow is a deformation of rotationally symmetric Taylor vortex flow. Increase Re Flow is steady in co-moving frame Poincare Sections Dividing stream surface breaks up => particles can migrate from vortex to vortex
Methods • • • Solve Navier-Stokes equations numerically to obtain wavy vortex flow. Finite differences (MAC method); Pseudo-spectral (P. S. Marcus) 2. Integrate particle path equations (20, 000 particles) in a frame rotating with the wave (4 th order Runge-Kutta).
Wavy Vortex Flow Poincare Section near onset of waves 6 vortices 1 2 3 4 5 6 inner cylinder outer cylinder
At larger Reynolds numbers (Rudman, Metcalfe, Graham: 1998)
Effective Diffusion Coefficient Characterize the migration of particles from vortex to vortex Rudman, AICh. E J 44 (1998) 1015 -26. Initialization: Uniformly distribute 20, 000 particles Taylor vortices (dimensionless) Wavy vortices
Size of mixing region Dz (dimensionless)
An Eulerian Approach Symmetry Measures Theoretical Fact A three dimensional phase space is necessary for chaotic trajectories. The Idea (Mezic): Deviation from certain continuous symmetries can be used to measure the local deviation from 2 D For Wavy Vortex Flow rotational symmetry and dynamical symmetry : • If either is zero, then flow is locally integrable, so as a diagnostic we consider the product
Dynamical Symmetry Steady incompressible Navier-Stokes equations in the form Equation of motion for B B is a constant of the motion if
155 Reynolds Number 162 324 486 648
155 Reynolds Number 162 324 486 648
155 Reynolds Number 162 324 486 648 Looks interesting, but correlation does not look strong !
Averaged Symmetry Measures and partial averages
Size of chaotic region Dz D
Serendipity ! King, Rudman, Rowlands and Yannacopoulos Physics of Fluids 2000
Effect of Radius Ratio (Mind the Gap) Dz/ Re/Rec
Effect of Flow State : Axial wavelength m: Number of waves Re/Rec
Effect of Flow State Dz Re/Rec
Summary • Dz is highly correlated with < >< > • The correlation is not perfect. • The symmetry arguments are general • Yannacopoulos et al (Phys Fluids 14 2002) show that Melnikov function, M ~ < >. Is it good for anything else?
2 D Rotating Annulus u(r, z, t) Richard Keane’s results (see poster) Symmetry measure: FSLE Log(FSLE) Log(<|d /dt|>)
Prandtl-Batchelor Flows (Batchelor, JFM 1, 177 (1956) Steady Navier-Stokes equations in the form Integrating N-S equation around a closed streamline s yields
Break-up of Closed Streamlines Yannacopoulos et al, Phys Fluids 14 2002 (see also Mezic JFM 2001) Expand This is the Melnikov function